Evaluate the following expression. Your answer must be exact. $\left(\dfrac{9\sqrt{2}}{2}+\dfrac{9\sqrt{2}i}{2}\right)^2=$
Answer: The Strategy The easiest way to find $z^{n}$ for a complex number $z=({a}+{b}i)$ is using its modulus and argument. Therefore, our solution will consist of the following steps: Find the modulus and argument of $z$. [How is this done, in general?] Find the modulus and argument of $z^{n}$. [How is this done, in general?] Find the rectangular form $z^{n}$. Find the modulus and argument of $\left(\dfrac{9\sqrt{2}}{2}+\dfrac{9\sqrt{2}}{2}i\right)$ $\left({\dfrac{9\sqrt{2}}{2}}+{\dfrac{9\sqrt{2}}{2}}i\right)$ is of the form $({a}+{b}i)$, where ${a=\dfrac{9\sqrt{2}}{2}}$ and ${b=\dfrac{9\sqrt{2}}{2}}$. Therefore: $\begin{aligned}r&=\sqrt{{a}^2 + {b}^2} \\\\&=\sqrt{ \left({\dfrac{9\sqrt{2}}{2}}\right)^2 + \left({\dfrac{9\sqrt{2}}{2}}\right)^2} \\\\&=\sqrt{{\dfrac{81}{2}}+{\dfrac{81}{2}}} \\\\&=9\end{aligned}$ Using the arctangent formula, we have: $\begin{aligned}\theta&=\arctan\left(\dfrac{{b}}{{a}}\right) \\\\&=\arctan\left(\dfrac{{\dfrac{9\sqrt{2}}{2}}}{{\dfrac{9\sqrt{2}}{2}}}\right) \\\\&=45^\circ\end{aligned}$ Since both ${a=\dfrac{9\sqrt{2}}{2}}$ and ${b={\dfrac{9\sqrt{2}}{2}}}$ is positive, $\left(\dfrac{9\sqrt{2}}{2}+\dfrac{9\sqrt{2}}{2}i\right)$ lies in Quadrant $1$. Therefore, $\theta$ must be between $0^\circ$ and $90^\circ$, so our answer matches our requirements. Find the modulus and argument of $\left(\dfrac{9\sqrt{2}}{2}+\dfrac{9\sqrt{2}i}{2}\right)^2$ We found that the modulus and argument of $\left({\dfrac{9\sqrt{2}}{2}}+{\dfrac{9\sqrt{2}}{2}}i\right)$ are $9$ and $45^\circ$. Therefore, the modulus and argument of $\left({\dfrac{9\sqrt{2}}{2}}+{\dfrac{9\sqrt{2}}{2}}i\right)^2$ are $9^2=81$ and $(45^\circ)\cdot2=90^\circ$. Find the rectangular form of $\left(\dfrac{9\sqrt{2}}{2}+\dfrac{9\sqrt{2}i}{2}\right)^2$ Since the argument is $90°$, we know the number lies on the positive side of the imaginary number axis and is therefore a positive pure imaginary number. Since the modulus is $81$, our solution is $81i$. [What does this look like graphically?] [How do we find this algebraically?] Summary $\left(\dfrac{9\sqrt{2}}{2}+\dfrac{9\sqrt{2}}{2}i\right)^2=81i$